Compressed sensing with side information: Geometrical interpretation and performance bounds

Abstract

We address the problem of Compressed Sensing (CS) with side information. Namely, when reconstructing a target CS signal, we assume access to a similar signal. This additional knowledge, the side information, is integrated into CS via ℓ1-ℓ1 and ℓ1-ℓ2 minimization. We then provide lower bounds on the number of measurements that these problems require for successful reconstruction of the target signal. If the side information has good quality, the number of measurements is significantly reduced via ℓ1-ℓ1 minimization, but not so much via ℓ1-ℓ2 minimization. We provide geometrical interpretations and experimental results illustrating our findings.Index Terms-Compressed sensing, basis pursuit, ℓ1-ℓ1 minimization, ℓ1-ℓ2 minimization, Gaussian width.

Dynamic sparse state estimation using &#x2113;<inf>1</inf>-&#x2113;<inf>1</inf> minimization: Adaptive-rate measurement bounds, algorithms and applications

“…First, in practice, it is easier to obtain bounds on the magnitude of η[k] than it is to tune the parameter β2. Second, the recent results in [8,9] establish reconstruction guarantees for (2) in the case of static signals; those results also establish an optimal value for the parameter β (equal to 1), making (2) parameter-free.…”

“…where Si is the event "perfect reconstruction at time i" and Ei is the event in (9). Simple algebraic manipulation shows that if we replace the expression for mi−1 (in step 17) in (9) , we obtain…”

“…From (13) to (14), we used the fact that Si|E = Si|Ei, for 3 ≤ i ≤ k, and that the events Si conditioned on Ei (i.e., (9)) are independent, for 3 ≤ i ≤ k. Now note that, since mi ≥ m and…”

“…Note that, in general, (2) may have more than one solution. Based on the results in [8,9], we propose a recursive mechanism to compute the number of measurements m k at each time k. This scheme minimizes m k while guaranteeing perfect reconstruction in the noiseless scenario, η[k] = 0, or stable reconstruction (i.e., x[k] − x[k] 2 ≤ 2σ k /ǫ, for some 0 < ǫ < 1) in the noisy scenario. Furthermore, note that there are no parameters (weights) to tune in (2).…”

Reference-based compressed sensing: A sample complexity approach

“…Applying it requires three elements: a basis in which the signals are sparse, an acquisition matrix with specific properties, and a nonlinear procedure to reconstruct signals from their measurements, e.g., 1-norm minimization. After the initial work [1,2], much research focused on reducing acquisition rates even further, by leveraging more structured signal information [4][5][6][7][8], using prior information [9][10][11][12][13][14][15][16][17], or improving reconstruction algorithms, e.g., via reweighting schemes [18][19][20][21][22].…”